Note that:
\(\left( x+y\right) ^{2}-2xy= x^{2}+y^{2}\)
\(\left( x-y\right) ^{2}+2xy= x^{2}+y^{2}\)
\(\left(x-y\right) ^{3}+3xy\left( x-y\right) =x^{3}-y^{3}\)
\(\left( x+y\right) ^{3}-3xy\left( x+y\right) =x^3 + y^{3}\)
\(\left( x+y+z\right) ^{2}-2\left( xy+yz+xz\right) =x^{2}+y^{2}+z^{2}\)
Applicable questions:
Question 1: If x + y = a and xy = b, what is the sum of the reciprocals of x and y?
Solution:
\(\dfrac {1} {x }+\dfrac {1} {y}=\dfrac {x +y} {xy}\)= \(\dfrac {a} {b}\)
Question 2: If \(x^{2}+y^{2}=153\) and x + y = 15, what is xy?
Solution:
\(\left( x+y\right) ^{2}-2xy= x^{2}+y^{2}\)
\(15^{2}-2xy=153\)\(\rightarrow xy=36\)
Question 3: If \(\left( x+y\right) ^{2}=1024\) , \(x^{2}+y^{2}\) = 530 and x > y , what is x - y?
Solution:
\(\left( x+y\right) ^{2}-2xy=x^{2}+y^{2}\)
1024 - 2xy = 530\(\rightarrow 2xy=494\)
\(\left( x-y\right) ^{2}+2xy=x^{2}+y^{2}\)
\(\left( x-y\right) ^{2}=36\)
x - y = 6
Question 4: x + y = 3 and \(x^{2}+y^{2}=89\), what is \(x^{3}+y^{3}\)?
Solution:
\(\left( x +y\right) ^{2}-2xy=x^{2}+y^{2}\)
9 - 2xy = 89 \(\rightarrow -2xy=80\) so xy = -40
\(\left( x+y\right) ^{3}-3xy\left( x+y\right) =27 - 3(-40)* 3 =
27 + 3*40*3 = x ^{3}+y^{3}\)
\(x ^{3}+y^{3}\)= 387
Question #5: If \(x+\dfrac {1} {x}=5\), what is \(x^{3}+\dfrac {1} {x ^{3}}\)?
Solution:
\(\left( x+\dfrac {1} {x}\right) ^{3}=x^{3}+3x^{2}.\dfrac {1} {x}+3x.\dfrac {1} {x^{2}}+\dfrac {1} {x^{3}}\)
\(5^{3}=x^{3}+3\left( x+\dfrac {1} {x}\right) +\dfrac {1} {x^{3}}\)
125 - 3*5 = \(x^{3}+\dfrac {1} {x ^{3}}\)
The answer is 110.
Question #6 : 2011 Mathcounts state sprint #24 : x + y + z = 7 and \(x^{2}+y^{2}+z^{2}=19\), what is the arithmetic mean of the three
product xy + yz + xz?
Solution:
\(\left( x+y+z\right) ^{2}-2\left( xy+yz+xz\right) =x^{2}+y^{2}+z^{2}\)
\(7^{2}-2\left( xy+yz+xz\right) =19\)
xy + yz + xz = 15 so their mean is \(\dfrac {15} {3}=5\)
More practice problems (answer key below):
#1:If x + y = 5 and xy = 3, find the value of \(\dfrac {1} {x^{2}}+\dfrac {1} {y^{2}}\).
#2: If x + y = 3 and \(x^{2}+y^{2}=6\), what is the value of \(x^{3}+y^{3}\)?
#3: The sum of two numbers is 2. The product of the same two numbers is 5.
Find the sum of the reciprocals of these two numbers, and express it in simplest form.
#4:If \(x-\dfrac {6} {x}\) = 11, find the value of \(x^{3}-\dfrac {216} {x^{3}}\)?
#5: If \(x+\dfrac {3} {x} = 9\), find the value of \(x^{3}+\dfrac {27} {x^{3}}\)?
#6:If \(x+\dfrac {1} {x} = 8\), what is \(x^{3}+\dfrac {1} {x ^{3}}\)?
Answers:
#1 :\(\dfrac {19} {9}\)
#2: 13.5
#3: \(\dfrac {2} {5}\)
#4: 1529 [ \(11^{3}\)+ 3 x 6 x 11 =1529]
#5: 648 [\(9^{3}\)-3 x 3 x 9 = 648]
#6: 488 [ \(8^{3}\)– 3 x 8 = 488]
Showing posts with label algebra manipulation. Show all posts
Showing posts with label algebra manipulation. Show all posts
Wednesday, July 6, 2016
Tuesday, April 16, 2013
This Week's Work : Week 8 -- for Inquisitive Young Mathletes
Assignment 1:
Using Algebra and Number Sense as Shortcuts from Mathcounts Mini
Watch the video and work on the activity sheet below the video on the same link for more practices.
Also, review the following:
\(x^{2}-y^{2}=\left( x+y\right) \left( x-y\right)\) \(\left( x+y\right) ^{2}=x^{2}+2xy+y^{2}\) \(\left( x-y\right) ^{2}=x^{2}-2xy+y^{2}\) \(\left( x+y\right) ^{2}-2xy =x^{2}+ y^{2}\)
\(\left( x-y\right) ^{2}+ 2xy =x^{2}+y^{2}\)
\(\left( x+y\right) ^{2}-4xy =\left( x-y\right) ^{2}\)
Assignment 2:
Pascal's Triangle from Math is Fun
Pascal's Triangle and Its Patterns
Assignment 3:
It'll be sent through e-mail.
Happy problem solving !!
Using Algebra and Number Sense as Shortcuts from Mathcounts Mini
Watch the video and work on the activity sheet below the video on the same link for more practices.
Also, review the following:
\(x^{2}-y^{2}=\left( x+y\right) \left( x-y\right)\) \(\left( x+y\right) ^{2}=x^{2}+2xy+y^{2}\) \(\left( x-y\right) ^{2}=x^{2}-2xy+y^{2}\) \(\left( x+y\right) ^{2}-2xy =x^{2}+ y^{2}\)
\(\left( x-y\right) ^{2}+ 2xy =x^{2}+y^{2}\)
\(\left( x+y\right) ^{2}-4xy =\left( x-y\right) ^{2}\)
Assignment 2:
Pascal's Triangle from Math is Fun
Pascal's Triangle and Its Patterns
Assignment 3:
It'll be sent through e-mail.
Happy problem solving !!
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